Mathematics and Computation (2019) [pdf]
math.ias.edu
math.ias.edu
Roughly it accepts inputs that have at least 2/3rds of witnesses accepting and rejects inputs that have no more than 1/3 of witnesses accepting. Witness means additional input (usually considered random input). The super nicety is the huge gap between 1/3 and 2/3.
One can simulate a BPP recognizer to a high degree of fidelity. Just try a bunch of random witnesses.
However, we don't yet know how to efficiently perfectly implement a perfect recognizer. Until we have sampled a lot of witnesses we really don't know what fraction the of overall population we are drawing from is accepting.
However (as the book points out) we know the strategy for perfect solution. We can decide BPP perfectly and efficiently if and only if certain very strong efficient pseudo random number generators exist. And the existence of such is very much tied to if certain problems are hard (require large circuits to solve) or not.
The book should be called Mathematics and Theory of computation
Modern TCS is about unifying a lot of the ad-hoc approaches of old, as well as analyzing different models of computation that better model reality (EMM, streaming, distributed, etc).
I like both.